428 XV NORMED ALGEBRAS AND SPECTRAL THEORY
B = (7+ 7T*)-1 and if for each y e E we put x = B • y, show that the hypotheses
imply that y e dom(T*), by using the fact that x e dom(r*).)
12. Let TV be a normal operator on E. If dom(Nz) — dom(N), show that N is continuous.
(If not, there would exist a closed subspace F of E, such that F is the Hilbert sum of
an infinite sequence (Fn) of nonzero subspaces stable under N and N*, such that the
restriction Nn of N to Fn is a continuous operator and such that
for xn e Fn , where the sequence (an) is increasing and tends to +00.)
13. EXTENSIONS OF HERMITIAN OPERATORS
(15.13,1) Let E be a separable Hilbert space. A not necessarily bounded
operator H on E is said to be hermitian if: (1) dom(H) is dense in E; (2)
dom(H) c domCff*), and the restriction of H* to dom(H) is equal to H; in
other words, for all x, y in dom(H)9
(15.13.1.1) (H-x\y) = (x\H-y).
In particular, (H • x \ x) is real for all x e dom(H).
It should be noted that in general H* is not hermitian. When H is con-
tinuous, this definition agrees with that of (11.5), and a continuous hermitian
operator is the same thing as a continuous self-adjoint operator. An unbounded
self-adjoint operator is always hermitian, but as we shall see below there exist
closed hermitian operators which are not self-adjoint.
Example
(15.13.2) Let Q) be the vector space of indefinitely differentiable complex-
valued functions on R with compact support. If /? denotes Lebesgue measure
on R, then @ may be identified with a subspace of L£(R, ft) (13.19), which is
dense in L£(R, /?) (17.1.2). It follows immediately from the formula for
integration by parts (8.7.5) and the fact that the functions belonging to ^
vanish outside a compact set, that for all x9 y in ^
= f Dx(t)XT) dt=- f x(ODXF) dt = -(x \ Dy).
(D* | y)
This may be expressed by saying that iD = H is a hermitian operator. The
conclusion is the same if we replace L£(R, ft) by Lc(I, /?), where I is a (boundedrem and the